Upper and lower bounds for the iterates of order-preserving homogeneous maps on cones

We define upper bound and lower bounds for order-preserving homogeneous of degree one maps on a proper closed cone in ${\mathbb{R}}^n$ in terms of the cone spectral radius. We also define weak upper and lower bounds for these maps. For a proper closed cone $C\subset {\mathbb{R}}^n$, we prove that any order-preserving homogeneous of degree one map $f:\text{inter}C\to \text{inter}C$ has a lower bound. If $C$ is polyhedral, we prove that the map $f$ has a weak upper bound. We give examples of weak upper bounds for certain order-preserving homogeneous of degree one maps defined on the interior of ${\mathbb{R}}_{+}^n$.